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In this article, we discuss the problem about the properties on solutions for several types of q-difference equations and obtain some results on the exceptional values of transcendental meromorphic solutions fz with zero order, their q-differences Δqfz=fqz−fz, and divided differences Δqfz/fz. In addition, we also investigated the condition on the existence of rational solution for a class of q-difference equations. Our theorems are some extensions and supplement to those results given by Liu and Zhang and Qi and Yang.

In this paper, we investigate the growth of meromorphic solutions of some kind of non-homogeneous linear difference equations with special meromorphic coefficients. When there are more than one coefficient having the same maximal order and the same maximal type, the estimates on the lower bound of the order of meromorphic solutions of the involved equations are obtained. Meanwhile, the above estimates are sharpened by combining the relative results of the corresponding homogeneous linear difference equations.

The purpose of this manuscript is to study some properties on meromorphic solutions for several types of q-difference equations. Some exponents of convergence of zeros, poles and fixed points related to meromorphic solutions for some q-difference equations are obtained. Our theorems are some extension and improvements to those results given by Qi, Peng, Chen, and Zhang.

In this paper, we investigate the relationships between fixed points of meromorphic functions, and their higher order differences and shifts, and generalize the case of fixed points into the more general case for first order difference and shift. Concretely, some estimates on the order and the exponents of convergence of special points of meromorphic functions and their differences and shifts are obtained.

The purpose of this article is to investigate the growth of meromorphic solutions for the system of complex functional equations with a more general form. Some estimates of the growth order of its meromorphic solutions in all possible cases are obtained, and several examples show our estimates are sharp to some certain.

The main purpose of this article is to investigate some properties on the meromorphic solutions of some types of q-difference equations, which can be seen the q-difference analogues of Paineve equations. We obtain estimates of the exponent of convergence of poles of $\Delta_qf(z):=f(qz)-f(z)$, which extends some earlier results by Chen et al.

For a transcendental meromorphic function f ( z ) , the main aim of this paper is to investigate the properties on the zeros and deficiencies of some differential-difference polynomials. Some results about the deficiencies of some differential-difference polynomials concerning Nevanlinna deficiency and Valiron deficiency are obtained, which are a generalization of and improvement on previous theorems given by Liu, Lan and Zheng, etc.

In this paper, we study the relation between the deficiencies concerning a meromorphic function f(z), its derivative f′(z) and differential-difference monomials f(z) m f(z+c)f′(z), f(z+c) n f′(z), f(z) m f(z+c). The main results of this paper are listed as follows: Let f(z) be a meromorphic function of finite order satisfying lim sup r → + ∞ T ( r , f ) T ( r , f ′ ) < + ∞ , $$\mathop {\lim \,{\rm sup}}\limits_{r \to + \infty } {{T(r,\,f)} \over {T(r,\,f')}}{\rm{ < }} + \infty ,$$ and c be a non-zero complex constant, then δ(∞, f(z) m f(z+c)f′(z))≥δ(∞, f′) and δ(∞,f(z+c) n f′(z))≥ δ(∞, f′). We also investigate the value distribution of some differential-difference polynomials taking small function a(z) with respect to f(z).

In this paper, we investigate the value distribution of meromorphic solutions of homogeneous and non-homogeneous complex linear differential-difference equations, and obtain the results on the relations between the order of the solutions and the convergence exponents of the zeros, poles, a-points and small function value points of the solutions, which show the relations in the case of non-homogeneous equations are sharper than the ones in the case of homogeneous equations.

In this article, we study the uniqueness problem of meromorphic functions in m-punctured complex plane Ω and obtain that there exist two sets S 1, S 2 with ♯S 1 = 2 and ♯S 2 = 9, such that any two admissible meromorphic functions f and g in Ω must be identical if f, g share S 1, S 2 I M in Ω.

The purpose of this paper is to investigate the fixed points of solutionsf(z)of someq-difference equations and obtain some results about the exponents of convergence of fixed points off(z)andf(qjz) (j∈N+),q-differencesΔqf(z)=f(qz)-f(z), andq-divided differencesΔqf(z)/f(z).

The characteristic functions of differential-difference polynomials are investigated, and the result can be viewed as a differential-difference analogue of the classic Valiron-Mokhon’ko Theorem in some sense and applied to investigate the deficiencies of some homogeneous or nonhomogeneous differential-difference polynomials. Some special differential-difference polynomials are also investigated and these results on the value distribution can be viewed as differential-difference analogues of some classic results of Hayman and Yang. Examples are given to illustrate our results at the end of this paper.

In this paper, we continue to investigate the value distribution of some difference polynomials, which can be considered as difference counterparts of some classic results of Hayman.

In this paper, the authors continue to study the growth of meromorphic solutions of homogeneous or non-homogeneous linear difference equations with entire coefficients, and obtain some results which are improvement and extension of previous results in Chiang and Feng (2008) [7] and Laine and Yang (2007) [19] . Examples are also given to illustrate the sharpness of our results.

We investigate higher order difference equations and obtain some results on the growth of transcendental meromorphic solutions, which are complementary to the previous results. Examples are also given to show the sharpness of these results. We also investigate the growth of transcendental entire solutions of a homogeneous algebraic difference equation by using the difference analogue of Wiman-Valiron Theory.

We investigate the growth of transcendental meromorphic solutions of some complex q-difference equations and find lower bounds for Nevanlinna lower order for meromorphic solutions of such equations. We also obtain a q-difference version of Tumura–Clunie theorem.

In this paper, we investigate the growth of two composite entire functions of finite iterated order and a series of comparative growths of log p + q T ( r , f ( g ) ) ( p , q ∈ N ) with log p T ( r , f ) and log q T ( r , g ) . At the end of this paper, we apply some growth results into the factorization of the solutions of linear differential equation. We achieve some results which are the improvements and extensions of the previous results.

In this paper, we study the value distribution of a meromorphic function concerning its derivative and q-shift difference , where is of finite logarithmic order. We also investigate the uniqueness of differential-q-shift-difference polynomials with more general forms of entire functions of order zero.

In this paper, we investigate the growth of meromorphic solutions to a complex higher order linear differential equation whose coefficients are meromorphic functions of $[p,q]$-orders. We get the results about the lower $[p,q]$-order of solutions of the equation. Moreover, we investigate the $[p,q]$-convergence exponent and the lower $[p,q]$-convergence exponent of distinct zeros of $f(z)-varphi(z).$

In this paper, we deal with the relationship between the small function and the derivative of solutions of higher order linear differential equations

In this paper, we investigate the growth and the exponent of convergence of the sequence of φ-points of meromorphic solutions of the linear differential equations A k ( z ) f ( k ) + A k − 1 ( z ) f ( k − 1 ) + ⋯ + A 1 ( z ) f ′ + A 0 ( z ) f = 0 $$A_{k}(z)f^{(k)}+A_{k-1}(z)f^{(k-1)}+ \cdots+A_{1}(z)f'+A_{0}(z)f=0 $$ and A k ( z ) f ( k ) + A k − 1 ( z ) f ( k − 1 ) + ⋯ + A 1 ( z ) f ′ + A 0 ( z ) f = F ( z ) , $$A_{k}(z)f^{(k)}+A_{k-1}(z)f^{(k-1)}+ \cdots+A_{1}(z)f'+A_{0}(z)f=F(z), $$ with entire coefficients A j ( z ) $A_{j}(z)$ , j = 0 , 1 , … , k $j=0,1,\ldots,k$ and F ( z ) $F(z)$ , where k ≥ 2 $k\geq2$ , A 0 ( z ) A k ( z ) ≢ 0 $A_{0}(z)A_{k}(z)\not\equiv0$ , φ ( z ) $\varphi(z)$ is a meromorphic function of finite order, and there is only one dominant coefficient A k ( z ) $A_{k}(z)$ of the maximal order, which is also a Fabry gap series.

In this paper, we investigate the growth and value distribution of solutions of some kinds of linear difference equations, where there may be more than one coefficient having the same maximal order and the same maximal type.

In this paper, we investigate the value distribution of a meromorphic function and its derivative concerning small functions in an angular domain and obtain some inequalities for a meromorphic function in an angular domain, which improve the previous results.

In this paper, we investigate the existence of transcendental meromorphic solutions of order zero of some nonlinear q-difference equations and some more general equations. We also give some estimates on the growth of transcendental meromorphic solutions of these equations. Some examples are given to illustrate the sharpness of some of our results.